# Construction of a global solution for the one dimensional singularly-perturbed boundary value problem

## TL;DR

This paper develops an $ ext{ε}$-uniform convergent approximate solution for a one-dimensional semilinear singularly-perturbed boundary value problem, improving accuracy through a repair process and confirming results with numerical experiments.

## Contribution

It introduces a novel method for constructing an $ ext{ε}$-uniform solution with enhanced convergence order using Green's function and a repair technique.

## Key findings

- Achieved $ ext{ε}$-uniform convergence of order $ ext{O}(N^{-1})$
- Improved to $ ext{O}( ext{ln}^2 N / N^2)$ after repair
- Numerical experiments confirm theoretical convergence rates

## Abstract

We consider an approximate solution for the one-dimensional semilinear singularly-perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an $\varepsilon$-uniform convergence of such gained the approximate solutions, in the maximum norm of the order $\mathcal{O}\left(N^{-1}\right)$ on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has $\varepsilon$--uniform convergence, but now of order $\mathcal{O}\left(\ln^2N/N^2\right)$ on $[0,1].$ In the end a numerical experiment is presented to confirm previously shown theoretical results.